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Chapter 2: Problem 110

Find the absolute maximum and minimum values of each function, and sketch the graph. $$F(x)=\left\\{\begin{array}{ll} x^{2}+4, & \text { for }-2 \leq x<0 \\ 4-x, & \text { for } 0 \leq x<3 \\ \sqrt{x-2}, & \text { for } 3 \leq x \leq 67 \end{array}\right.$$

Short Answer

Expert verified
Absolute maximum is approximately 8.06 at \(x = 67\). Absolute minimum is 1 at \(x = 3\).

Step by step solution

01

- Determine the intervals for each piece of the function

Identify the intervals over which each piece of the given piecewise function is defined. They are:1. \(-2 \leq x < 0 \) for \(x^{2} + 4\)2. \(0 \leq x < 3 \) for \(4 - x\)3. \(3 \leq x \leq 67 \) for \(\sqrt{x - 2}\)
02

- Evaluate the endpoints of each interval

To find the absolute maximum and minimum values, evaluate the function at the endpoints of each interval. Check each interval:1. \(F(-2)\) for \(x^{2} + 4\)2. \(F(0)\), \(F(3)\) for \(4 - x\)3. \(F(3)\), \(F(67)\) for \(\sqrt{x - 2}\)
03

- Calculate the function values at endpoints

Calculate the values:1. For \(x = -2\): \(F(-2) = (-2)^{2} + 4 = 4 + 4 = 8\)2. For \(x = 0\): \(F(0) = 4 - 0 = 4\)3. For \(x = 3 \) in both pieces:\(F(3)\) for \(4 - x = 4 - 3 = 1\)\(F(3)\) for \(\sqrt{x - 2} = \sqrt{3 - 2} = 1\)4. For \(x = 67\): \(F(67) = \sqrt{67 - 2} = \sqrt{65}\)
04

- Compare all the values

Compare all the calculated values to determine the absolute maximum and minimum points:1. \(F(-2) = 8\)2. \(F(0) = 4\)3. \(F(3) = 1\) from both piecewise segments4. \(F(67) = \sqrt{65}\) (approximately 8.06)Thus, the absolute maximum is approximately 8.06 at \(x = 67\), and the absolute minimum is 1 at \(x = 3\).
05

- Sketch the graph

Plot the graph according to the piecewise definitions:1. For \(-2 \leq x < 0\), sketch the parabola segment \(y = x^2 + 4\) from \(x = -2\) to close to \(x = 0\), undetermined at \(x=0\).2. For \(0 \leq x < 3\), sketch the line \(y = 4 - x\) from \(x = 0\) to close to \(x = 3\), connecting to the point at \(x=0\).3. For \(3 \leq x \leq 67\), sketch the segment \(y = \sqrt{x - 2}\) from \(x = 3\) to \(x = 67\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Maximum and Minimum
In a given function, the absolute maximum and minimum values represent the highest and lowest function values within a specific interval. For a piecewise function like in the given exercise, you need to evaluate each segment of the function separately. This involves:
  • Identifying all intervals and their endpoints.
  • Evaluating the function at these endpoints.
  • Comparing these values to find where the highest (maximum) and lowest (minimum) values occur.
In the provided exercise, the absolute maximum value is approximately 8.06 at the endpoint where \( x = 67 \). The absolute minimum value is 1, occurring at \( x = 3 \) for both segments of the piecewise function.
Interval Evaluation
Evaluating a piecewise function involves breaking it down into its defined intervals. A piecewise function can have different expressions depending on the range of input values. To work on this efficiently:
  • Identify intervals by analyzing each piece of the function. For our example: \(-2 \leq x < 0\), \(0 \leq x < 3\), and \(3 \leq x \leq 67\).
  • Calculate the function value at the endpoints of these intervals= \(x = -2, 0, 3, 67\).
  • Analyze intermediate points if necessary, but mainly focus on endpoints for maximum and minimum evaluations.
In practice, this helps ensure all key points are considered when determining function values across different segments.
Piecewise Functions
A piecewise function is defined using different expressions for different intervals of the domain. Each 'piece' has its own formula and range within the overall function. Understanding piecewise functions includes:
  • Breaking down each segment and its corresponding expression.
  • Knowing the domain restrictions for each segment, such as \(-2 \leq x < 0\) for the segment \(x^2 + 4\).
  • Recognizing the impact of each piece when evaluating limits, continuity, and differentiability.
Piecewise functions are especially useful to model situations where behavior changes at certain points, like tax brackets or shipping rates.
Function Graphing
Graphing a piecewise function involves plotting each segment on the same set of axes. This requires:
  • Accurately drawing each segment according to its specific formula and interval.
  • Ensuring transitions between segments align properly, based on endpoint values.
  • Maintaining clarity on whether endpoints are included (closed circles) or excluded (open circles).
In the exercise given, the graph includes:
  • A parabola from \(x = -2\) to close to \(x = 0\).
  • A linear segment from \(x = 0\) to \(x = 3\).
  • A square root function from \(x = 3\) to \(x = 67\).
Combining these accurately on one graph provides a clear visual representation of the piecewise function.

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Most popular questions from this chapter

Find the absolute maximum and minimum values of each function, and sketch the graph. $$h(x)=\left\\{\begin{array}{ll} 1-x^{2}, & \text { for }-4 \leq x<0 \\ 1-x, & \text { for } 0 \leq x<1 \\ x-1, & \text { for } 1 \leq x \leq 2 \end{array}\right.$$

For what positive number is the sum of its reciprocal and four times its square a minimum?

Assume the function \(f\) is differentiable over the interval \((-\infty, \infty)\) : that is, it is smooth and continuous for all real numbers \(x\) and has no corners or vertical tangents. Classify each of the following statements as cither true or false. If you choose false, explain why. The function \(f\) can have no extrema but can have at least one point of inflection.

Consider the piecewise-defined function \(f\) defined by: \(f(x)=\left\\{\begin{array}{ll}x^{2}+2, & \text { for }-2 \leq x \leq 0 \\ 2, & \text { for } 0

$$\text {Graph each function using a calculator, iPlot, or Graphicus.}$$ $$f(x)=x^{2}+\frac{1}{x^{2}}$$
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